Multiple Integrals and Probability Notes for Math 2605

Transcription

1 Multiple Integrals and Probability Notes for Math 605 A. D. Andrew November 00. Introduction In these brief notes we introduce some ideas from probability, and relate them to multiple integration. Thus you will get practice evaluating multiple integrals and a preview of an important application that you will study in greater depth in the very near future.. Random Variables and Density Functions A random variable is the result of a random experiment. Some examples are a. The result X of choosing a real number at random from the interval [a,b]. The space of the random variable X is the set of possible outcomes, and in this case is the interval S [a,b]. Note that we're speaking of real numbers, so things such as p, as well as,, and, are possible outcomes. b. The amount of time Y that Professor Andrew spends fishing before catching his first trout. The space of Y is S [0, ). c. The SAT verbal score Z of a randomly chosen 6 year old. We will make the assumption that all scores are possible so that the space S of Z is the set of all real numbers. Each random variable comes with a density function f, which can be used to compute the probability that the value of the random variable is in a given set. Here are the density functions in the three cases above. a. If we pick a real number between a and b 0, then the density function is if x 0 f ( x) 8 and the probability that X lies in the interval [5,7] is 0 otherwise 7 P ( 5 X 7) f ( x) dx. Note that the probability that X lies in an 5 4 interval I is simply one-eigth the length of I. This is called the uniform density on [,0]. b. A reasonable density function for the time Y Professor Andrew waits for his first - x fish is f ( x) e, and the probability that he waits at least three hours is

2 Multiple Integrals and Probability Page of 7 P( Y - x ) e dx e exponential density. -, which is about.05. This is an example of an c. For the SAT verbal score we will use the famous bell-curve, a Normal distribution with mean m 506 and standard deviation s. We will explain the terms mean and standard deviation shortly. This density function is Ê ( x - 506) ˆ f (x) expá - ( p )() Ë (), and the probability that the random 550 student scores between 50 and 550 is f ( t) dt ª.4. This last integral cannot be computed exactly, but it is so useful that its values are widely tabulated. The basic properties of probabilities are a. ( X Œ A ) 0 P for every set A, b. P( X A» B ) P( X Œ A ) + P( X Œ B ) c. ( S ) 50 Œ whenever A and B are disjoint, P.. Pairs of Random Variables When we deal with pairs of random variables X, Y, the density function is a function of two variables and is called the joint density function of X and Y. Here are a few examples. a. Let X and Y be random real numbers chosen independently from [0,] and consider the point (X,Y) in the unit square in the first quadrant of the x-y plane. The joint density function of X and Y is identically equal to on the unit square, and the probability that (X,Y) lies inside the quarter circle is p dx dy. {(, ) Œ : + } 4 x y S x y b. Let X and Y be random real numbers chosen independently from [0,], and let Z X + Y. The joint density function of X and Z is constantly equal to on a parallelogram with vertices at (0,0), (,), (,), and (0,), and the probability that Independent is a technical term, and we will not define it here. However, common sense dictates that two numbers chosen from [0,] may well be independent, but the SAT verbal and SAT math scores of a particular student are not. If X and Y are independent numbers in [0,] and Z is their sum, then X and Z are not independent.

3 Multiple Integrals and Probability Page of 7 Z > is + x x 0z dz dx. In the figure below we show the domain of the joint density function. The shaded portion of the parallelogram is the set where Z >. c. If X is the amount of time Professor Andrew waits before catching his first fish and Y is the amount of time Professor Wang waits before catching his first fish, then the joint density of X and Y is f (x, y) e - x - y. The probability that Professor Andrew catches his first fish before Professor Wang does is P( X < Y ) 4. Marginal Densities y - x - y e y 0 x 0 dx dy If we know the joint density function of X and Y, then we may compute the density function of X and the density function of Y. These are called marginal densities. The density function of X is. f X ( x) f ( x, y) dy and the density function of Y is f Y ( y) f ( x, y) dx. Visualize these in terms of parts of iterated integrals, and check for yourself that the density function for Professor Andrew's fishing time is - x - y f ( x) e and that for Professor Wang is f ( y) e. X 5. Mean and Variance The mean of a random variable X with density function g(x) is defined to be m X x g(x) dx. The variance of X is defined to be s X ( x - m X ) g(x) dx. You should think of the mean as the average value of X, and the variance as a measure of the spread of the distribution. For a pair of random variables X and Y, we can compute the mean and variance from the joint density function, or we may compute the marginal densities of X and Y and use them to find the mean of X and Y. That is Y

4 Multiple Integrals and Probability Page 4 of 7 () m X, m Y or ( ) ( ) x f (x,y) dx dy, y f (x, y) dx dy m Y y f Y ( y) dy. m X x f X ( x) dx, You should prove that these two methods produce the same result, and you should compare formula () with the discussion of centers of mass of plane regions in your calculus text. The variances of X and Y may also be computed from the marginal densities, as in s X ( x - m X ) f X (x) dx, but the covariance of X and Y can generally be computed only from the joint density function itself. The covariance measures the correlation between X and Y and is defined to be Cov(X,Y) ( X - m X )( Y - m Y ) dx dy. 6. An Example We now illustrate these ideas with one of the previous examples. Let X and Y be random real numbers chosen independently from [0,], and let Z X + Y. As described before, the joint density function f (x,z) of X and Z is constantly equal to on a parallelogram with vertices at (0,0), (,), (,), and (0,), shown below. To compute the probability that Z > X, we integrate the joint density above the z x diagonal

5 Multiple Integrals and Probability Page 5 of 7 P(Z > X) Ê + x ˆ Á dz dx Ë x 0 z x ( ( + x) - x) dx 0 - ( - x) 0 To find the marginal distribution of X, we simply integrate f (x,z) over a vertical line segment in this domain. That is f X (x) + x dz ( + x) - x, z x from which we see that X, not surprisingly, has uniform distribution on [0,]. To find the marginal distribution of Z, we integrate f (x,z) over a horizontal line segment in the domain. In this case, however, the endpoints of the line segment depend on the value of z. Specifically we see that f Z (z) z dx 0 z x 0 dx < z x - z z 0 z - z < z and the graph of f Z (z) is

6 Multiple Integrals and Probability Page 6 of 7 The shape reflects the fact that sums near are more likely than sums near 0 or. Please show that m X, s X, and m Z. Please also compute the variance of Z from the marginal distribution of Z. Here we will compute the variance of Z using the joint density s Z ( z - ) dxdz + x (z - ) dz dx x 0( z x ) x - ( x - ) x 0( ) dx 6 7. Exercises. In example 6 above, show that m X, s X, and m Z, calculate the variance of Z using the marginal distribution for Z, and calculate the covariance of X and Z. Calculate P( X < and Z > ) and P( X > and Z < 4 ).. Suppose the joint probability density function of X and Y is 0 x y 0 y x f ( x, y). 0 otherwise a. Sketch the support (set on which f is not zero) of f. b. Compute the marginal probability density functions f X (x) and f Y (y) Ê ˆ Ê ˆ c. Calculate P Á X and P Ë ÁY. Ë 4 Ê ˆ d. Calculate P Á X and Y. Ë 4 Ê X ˆ e. Calculate P ÁY. Ë. In the Andrew-Wang trout tournament, compute the probability that Andrew catches his first trout in three hours or less and Wang catches his first in two hours or less.

7 Multiple Integrals and Probability Page 7 of 7 Compute the probability that it Wang catches his first trout in less than half the time it takes Andrew to catch his first. 4. Let X and Y have joint probability density function f (x, y) on the set x y, 0 x. Calculate a. P( 0 X ) b. P( Y ) c. P( X and Y ) 5. Consider the problem.a. above, where points are chosen randomly in the unit square. Use a computer program of your choice to generate 0,000 random points in the square and compute the fraction of them that lie inside the quarter circle. Use this fraction to approximate p. 8. Selected Answers ( ). Cov(X,Z), P X < and Z > P X > and Z < ( 4) 8,. a. The triangle with vertices (0,0), (,0), (,). b. f X (x) 5 x 4, f Y (y) 0 y ( - y ) Ê c. P X ˆ Á Ë, P Ê Y ˆ Á Ë 4 5 Ê d. PÁ X and Y Ë 4 ˆ 0 Ê e. P Y X ˆ Á Ë 4 9. Reference Hogg and Tanis, Probability and Statistical Inference, 6th edition, Prentice Hall, 00.